# Subgraph Patterns: Network Motifs and Graphlets. Pedro Ribeiro

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1 Subgraph Patterns: Network Motifs and Graphlets Pedro Ribeiro

2 Analyzing Complex Networks We have been talking about extracting information from networks Some possible tasks: General Patterns Ex: scale-free, small-world Community Detection What groups of nodes are related? Node Classification Importance and function of a certain node? Network Comparison What is the type of the network? 2

3 Building Blocks of Networks Subnetworks, or subgraphs, are the building blocks of networks They have the power to characterize and discriminate networks 3

4 Building Blocks of Networks 4

5 Some Graph Definitions Subgraph: subset of vertices and edges inside another graph Induced subgraph: You must consider all connections between selected nodes Induced Subgraph Non-Induced Subgraph 5

6 Some Graph Definitions Graph Isomorphism: Two graphs G and H are isomorphic (G ~ H) when there is a one to one mapping between the nodes of both graphs and there is an edge between two vertices of G if and only if the corresponding vertices in H also form an edge Informally we are seeing if the topology is the same 4 Isomorphic Graphs 6

7 Some Graph Definitions Graph Automorphisms: Set of isomorphisms of a graph into itself Equivalence of Nodes Two nodes are equivalent if there exists an automorphism that maps one into the other This relation partitions a graph into equivalence classes Informally, we are looking at the graph symmetry Example graph automorphisms and equivalence classes 7

8 Some Graph Definitions Graph Matches: A match of a graph H in a larger graph G is a set of nodes of G that induce the respective subgraph H. In other words, a subgraph of G that is isomorphic to H 8

9 Example Application of Subgraphs Consider all possible directed subgraphs of size 3 9

10 Example Application of Subgraphs Z-Score measures overrepresentation Different networks present similar fingerprints Image: Adapted from (Milo et al., 2004) 10

11 Network Motifs Milo et al. (2002) came up with the definition of network motifs: recurring, significant patterns of interconnections How to define: Pattern: induced subgraph Recurring: found many times, i.e., high frequency Significant: more frequent than it would be expected in similar networks (same degree sequence) Parameters P, U, D, N control the definition (Milo et al., 2002, used {0.01, 4, 0.1, 1000}) Image: Adapted from (Milo et al., 2004) 11

12 Network Motifs Random Networks Motif Original Network Image: Adapted from (Milo et al., 2002) 12

13 Network Motifs: Example Application Table: Adapted from (Milo et al., 2002) 13

14 Network Motifs: Example Application Image: taken from (Sporns and Kotter, 2004) 14

15 Graphlet Degree Distribution What about a node subgraph metric? The degree distribution is in a way measuring participation in subgraphs of size 2 Can we generalize this? Przulj (2006) came up with the definition of graphlet degree distribution: Where does the node appear in orbits of subgraphs? 15

16 Graphlet Degree Distribution 16

17 Computational Problem In its core, finding motifs and graphlets its all about finding and counting subgraphs. Just knowing if a certain subgraph exists is already an hard computational problem! Subgraph isomorphism is NP-complete Execution time grows exponentially as the size of the graph or the motif/graphlet increases Feasible motif size is usually small (3 to 8) and network size in the order of hundreds or thousands of nodes 17

18 Subgraph Discovery Research Part of my research is focused on improving efficiency in network motif detection. Scale up! How? Novel data structures for the graphs and subgraphs Novel faster algorithms Sampling techniques Parallel approaches (with different paradigms) 18

19 Previous Approaches Network-centric approaches: Enumerate all subgraphs and then compute isomorphisms (ex: ESU/Fanmod, Kavosh). Subgraph-centric approaches: Find one subgraph at a time (ex: Grochow and Kellis) 19

20 A set-centric approach Key insight: can we do better looking for a given set of subgraphs? Looking for all does not allow for discarding subgraphs we don't care about (which would save computation time) Looking for one at the time does not allow to reuse results (which would avoid redundant computation) Can we find what is common between subgraphs and use that? Set-centric approach: Find a custom set of subgraphs (maybe one, maybe all, maybe something in between) 20

21 G-Tries Motivation Sequences and Prefix Trees (tries) Can the concept be extended to graphs? 21

22 G-Tries Motivation Graphs have common substructure! G-Tries (etimology: Graph retrieval ) 22

23 The G-Trie data structure G-Tries: (customized) collections of subgraphs Common substructures are identified Information is compressed 23

24 The G-Trie data structure G-Tries: also valid for directed networks 24

25 The G-Trie data structure G-Tries: also valid for colored/labeled networks G-Trie 25

26 Creating a G-Trie Iterative insertion Start with an empty g-trie 26

27 The Need for a Canonical Form There are different node orderings representing the same subgraph Canonical form for a getting an unique g-tries Different canon will give origin to different g-tries 27

28 Impact of Canonical Form 28

29 Custom Canonical Form Connectivity Path induces connected subgraph Compressibility More common subtructre, less g-tries nodes Constraining As many connections as possible to ancestor nodes (limit possible matches) GTCanon 29

30 GTCanon Example 30

31 Searching with G-Tries Backtracking Procedure Searching at the same time for several subgraphs Candidates for node 1: {0, 1, 2, 3, 4, 5} Try 0: Match = {0}, Neighb. = {1,3,4} Try 1: Match = {0,1}, Neighb. = {2,3,4,5} Try 2: no edge from 2 to 0! FAIL Try 3: no edge from 3 to 1! FAIL Try 4: Match = {0, 1, 4} FOUND! Try 5: no edge from 5 to 1! FAIL 31

32 Searching with G-Tries The same subgraph could be found several times due to automorphisms (symmetries) We would not only find {0,1,4} but also: {0, 4, 1} {1, 0, 4} {1, 4, 0} {4, 0, 1} {4, 1, 0} 32

33 Symmetry Breaking Conditions Conditions on node labels Add conditions to respective g-trie path Match only when conditions of at least one descendant are respected Filter conditions to ensure minimum work Ex: transitive property (a<b,a<c,b<c leads to a<b,b<c); assured descendants, only store relevant to node, etc 33

34 Complete G-Trie Example All six undirected 4-subgraphs 34

35 Complete G-Trie Example All thirteen directed 3-subgraphs 35

36 Sequential version: some results Set of 12 representative real networks Network Group Directed dolphins social no circuit physical no neural biological yes 297 2, metabolic biological yes 453 2, links social yes 1,490 19, coauthors social no 1,589 2, ppi biological no 2,361 6, odlis semantic yes 2,909 18, power physical no 4,941 6, company social yes 8,497 6, foldoc Semantic yes 13, , internet Physical no 22,963 48, ,390 V(G) 36 E(G) Nr. Neighbours Average Max

37 Sequential version: some results On both directed and undirected graphs g-tries were from 1 to 2 orders of magnitude faster than existing state of the art From 10x to 200x Example results for full census of size k (speedup on a set of undirected networks) Network k ESU Kavosh Grochow dolphins circuit coauthors ppi power internet

38 Sequential version: some results On both directed and undirected graphs g-tries were from 1 to 2 orders of magnitude faster than existing state of the art From 10x to 200x Example results for full census of size k (speedup on a set of directed networks) Network K ESU Kavosh Grochow neural metabolic links odlis company foldoc

39 Sequential version: some results On both directed and undirected graphs g-tries were from 1 to 2 orders of magnitude faster than existing state of the art From 10x to 200x 39

40 Sampling approach Sample some subgraph occurrences Deduce approximate total counting Trade accuracy for speed 40

41 Sampling approach: search tree Backtracking produces search tree 41

42 Sampling approach Original: unbalanced search tree Goal: Uniform sampling Subgraph occurrences are in last tree depth 42

43 Sampling approach Original: unbalanced search tree Goal: Uniform sampling Associate a probability of traversing with each search tree depth 100% 100% 80% 80% 43

44 Sampling Approach Probabilities associated with each depth: {P0, P1,, Pmax} Sampling is uniform Probability of any occurrence is P0 x P1 x x Pmax We can produce unbiased estimator Estimate = (Nr sampled occurences) / (P0 x P1 x x Pmax) The probabilities control the search Regarding accuracy: avoid small values close to root Entire search branches disregarded more variance Regarding speed: avoid high values close to root More search branches higher execution time Some results We can obtain ~90% of accuracy using ~20% of time 44

45 Parallelism Subgraph Discovery is a tree shaped computation Search nodes are independent from each other {0,1,3} If we know where we are, we can continue from there Tree Nodes -> Work Units 45

46 Parallel Problem Input: set of work units G-Trie: (network, g-trie node, partial match) Goal: efficiently distribute work units among processors We need dynamic load balancing 46

47 Parallel approaches Environments we explored Clusters (distributed memory using MPI) [almost linear speedup up to 128 cores] Multicores (shared memory using threads) [almost linear speedup up to 64 cores] Some of the techniques Receiver initiated work stealing Diagonal splitting and efficient snapshots Random polling 47

48 Example Computation G-Trie Node Graph Vertex 48

49 Example Computation G-Trie Node Graph Vertex 49

50 Example Computation G-Trie Node Graph Vertex 50

51 Example Computation G-Trie Node Graph Vertex 51

52 Example Computation G-Trie Node Graph Vertex Current Work Unit Explored Work Units STOP 52

53 Example Computation Keep Give to requester Both Diagonal Splitting 53

54 Some publications Core sequential algorithms - G-Tries: a data structure for storing and finding subgraphs. Data Mining and Know. Discovery, Towards a faster network-centric subgraph census. ASONAM' Querying Subgraph Sets with G-Tries. DBSocial'2012 (best paper award) - Strategies for Network Motifs Discovery. E-Science Sampling approach - Efficient Subgraph Frequency Estimation with G-Tries. WABI' Rand-FaSE: fast approximate subgraph census. SNAM, 2015 Parallel approach - Parallel subgraph counting for multicore architectures. ISPA' A Scalable Parallel Approach for Subgraph Census Computation. MuCoCos' Parallel Discovery of Network Motifs. Journal of Parallel and Distributed Computing Efficient Parallel Subgraph Counting using G-Tries. IEEE Cluster'2010. Concept variation and applications - Discovering Colored Network Motifs. Complenet' Motif Mining in Weighted Networks. Damnet'2012 (and new related in paper ACM-SAC'2015) - Co-authorship network comparison across research fields using motifs. ASONAM'

55 Software Reference sequential implementation is available online and uses C++: More versions available by request and we are working on giving more options to practitioners (ex: GUI, plugin for Gephi/Cytoscape) Graphlets, large scale and adaptive sampling almost public 55

56 Complex Networks "Group Pedro Ribeiro Fernando Silva (leader) David Aparício (PhD Student) Multicores, Graphlets Pedro Paredes (BSc Student) Network-Centric Graph Theory (leader) Miguel Araújo S. Choobdar (PhD Student) (former PhD Stud) Community Detection Matrix Factorization Structural Roles Temporal, Weigths Miguel Silva (MSc Student) Random Networks Ahmad Naser (MSc Student) GUI and Visualization Some collaborations (ongoing and/or starting): - M. Kaiser (U Newcastle) [Brain Networks] - L. F. Costa (U São Paulo) [Null Models] - K. Pingali (UT Austin) [Temporal Patterns] - C. Faloutsos (CMU) [Large Scale Networks] - S. Parthasarathy (Ohio State) [Roles] 56

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